Where two geodesic segments cross

The crossing of the geodesics through two segments on the ellipsoid, nearest the segments' middles, and whether it falls within both segments or out on their extensions.

Planning and education aid. Not for primary navigation. Full disclaimer

Computing your own values needs JavaScript and WebAssembly. The worked example below was computed when this page was built, and the documentation under "How we got this" applies.

53.373184847°

The geodesics cross at 53.373184847°, -14.563115348°: on both segments.

Crossing longitude
Within both segments
Where it falls
Along A
Along B
Length of A

Provenance

Computed by: navigation.geodesic.intersection 1.0.1, core 0.1.0
Model: Each segment's geodesic from its start at the inverse azimuth; the two crossings of the matching great circles seed Newton's method on the distances along both geodesics (to a nanometer); the crossing kept is the one nearest the segments' midpoints, |x − a/2| + |y − b/2|, as in GeographicLib's Intersect class
Accuracy: Matches GeographicLib IntersectTool -i on 60 random segment pairs: to a micrometer, or 5e-12 of the distance for crossings thousands of kilometers out at a shallow angle, where both are limited by rounding
Notes: None
Cites: Karney, C. F. F., Journal of Geodesy, Algorithms for geodesics; Karney, C. F. F., Journal of Surveying Engineering, Geodesic intersections

Something look off?

Your values

Showing an example. Change anything.

Segment A start latitude Decimal degrees, north positive, like 40.6413 · plain numbers mean degrees Segment A start longitude Decimal degrees, east positive, like -73.7781 · plain numbers mean degrees Segment A end latitude Decimal degrees, north positive, like 40.6413 · plain numbers mean degrees Segment A end longitude Decimal degrees, east positive, like -73.7781 · plain numbers mean degrees Segment B start latitude Decimal degrees, north positive, like 40.6413 · plain numbers mean degrees Segment B start longitude Decimal degrees, east positive, like -73.7781 · plain numbers mean degrees Segment B end latitude Decimal degrees, north positive, like 40.6413 · plain numbers mean degrees Segment B end longitude Decimal degrees, east positive, like -73.7781 · plain numbers mean degrees Ellipsoidoptional wgs84 (default), grs80, clarke1866, intl1924, airy1830, bessel1841, or krassovsky1940

More options 2

Custom semi-major axisoptional With inverse flattening, like 3396190 m (Mars) · plain numbers mean m Custom inverse flatteningoptional Like 169.894; 0 for a sphere

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How we got thisFormula, worked example, sources, and proof

Model: Each segment's geodesic from its start at the inverse azimuth; the two crossings of the matching great circles seed Newton's method on the distances along both geodesics (to a nanometer); the crossing kept is the one nearest the segments' midpoints, |x − a/2| + |y − b/2|, as in GeographicLib's Intersect class

Show your work

Distances to the crossing
Newton's method on the distances along A and B, from the great-circle crossings
A 5,554,908.791 m long at 51.381648°, B 2,952,243.683 m long at 157.189277° = 4,573,738.529 m along A, 1,271,766.763 m along B
Crossing latitude
direct problem along A
4,573,738.529 m from A's start = 53.373184847°

The same steps an agent gets from the MCP server with explain: true.

Accuracy: Matches GeographicLib IntersectTool -i on 60 random segment pairs: to a micrometer, or 5e-12 of the distance for crossings thousands of kilometers out at a shallow angle, where both are limited by rounding

When to use this: Use this to find where two great-circle routes cross: an airway against a boundary, a flight path against a corridor, a survey line against a parcel edge, or any two courses drawn between pairs of points. It says where they meet, how far along each the meeting lies, and whether it falls inside both segments or out beyond an end.

Limitations: Two geodesics on an ellipsoid meet twice, at antipodal points, and this returns the closer crossing, so a pair of very long routes can cross somewhere other than where you meant. A crossing beyond the end of a segment is still reported, with within saying no and the distance running past the segment's length, because knowing where two courses would meet if extended is often the question; it is not a claim that anything crosses. Segments that lie along the same geodesic have no single crossing. The edges are geodesics, so two rhumb lines drawn on a chart do not cross where this says.

Worked example: JFK to London and Reykjavík to Lisbon. Source: GeographicLib IntersectTool -i (Karney 2023).

You enter

Segment A end latitude: 51.47 deg
Segment A end longitude: -0.4543 deg
Segment A start latitude: 40.6413 deg
Segment A start longitude: -73.7781 deg
Segment B end latitude: 38.7223 deg
Segment B end longitude: -9.1393 deg
Segment B start latitude: 64.1466 deg
Segment B start longitude: -21.9426 deg

You get

Crossing latitude: 53.373184847°
Crossing longitude: -14.563115348°
Within both segments: yes
Where it falls: on both segments
Along A: 4,573,738.529 m
Along B: 1,271,766.763 m
Length of A: 5,554,908.791 m
Length of B: 2,952,243.683 m

Review: Not yet independently reviewed by a geodesist.

Last verified: 2026-09-18, when a maintainer last confirmed this tool's sources at the issuer. See the sources ledger.

Status: version 1.0.1, core 0.1.0. See this tool in the verification report.

Changes

2026-09-23, fixed: Short distances lost their meaning: a 0.859 m geodesic showed as 0.001 km. Distances in the navigation tools (version 1.0.1) still show to the meter in kilometers, but a short one now keeps three significant digits, down to the millimeter, so it shows as 0.000859 km. Differences such as the rhumb line's extra distance keep fixed decimals. Values are unchanged. Changelog

Checked against: 62 golden test vectors (download the test vectors, each with its source and tolerance). See how results are checked and every source.

Sources

Algorithms for geodesics, Karney, C. F. F., Journal of Geodesy, Vol. 87, No. 1. pp. 43-55.
Geodesic intersections, Karney, C. F. F., Journal of Surveying Engineering, Vol. 150, No. 3. 04024005 (closest intersection and segment intersections).